#545966

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(14,17)(18,20)(22,26)(24,27);(5,6,8,9)(7,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,28)(25,26);(1,3,4,2)(6,9)(10,11)(15,19)(18,20)(22,26)(23,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,23,25,28)(22,27,26,24);(5,7,8,12)(6,10,9,11)(13,14)(15,18)(16,17)(19,20)(21,24,25,27)(22,28,26,23)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 453162);